I need to rotate an object to meet another object (essentially, close a gap). I don't know what the angle is to do this and there doesn't appear to be a manipulator to just drag it around.

I attached a little sample diagram to illustrate. I want to rotate A on the circled point to bring it around and be flush to B. The lines may or may not be the same length, so I can't use point to point. This diagram is greatly simplified, in the actual case the objects are much more complex, the circled point would be more in the middle and closing the gap would swing the top out and create a gap at the top.

In the rotation dialog, click the button with a mouse pointer on it to define the rotation angle by mouse rather than entering a value. After clicking the OK button, you can click a reference point, then a target point. The rotation angle is the angle from (center - reference point) to (center - target point).

Thanks for this tip, I have been faced to that situation many times but I do not recall seeing that feature documented (I may have overlooked it). I always did a geometrical construction to measure my rotation angle and then apply the value in the box...